Question

The radioactive potassium- 40 isotope decays to argon-40 with a half-life of \(1.2 \times 10^{9}\) years. (a) Write a balanced equation for the reaction. (b) A sample of moon rock is found to contain 18 percent potassium-40 and 82 percent argon by mass. Calculate the age of the rock in years. (Assume that all the argon in the sample is the result of potassium decay.)

Short answer

The moon rock is approximately 2.05 billion years old.

Step-by-step solution

Write the Decay Equation

The decay of potassium-40 (\(_{19}^{40}K\)) into argon-40 (\(_{18}^{40}Ar\)) is a beta decay reaction. Using the atomic symbols, the balanced nuclear equation is: \[ \text{ }_{19}^{40}K \rightarrow \text{ }_{18}^{40}Ar + \beta^+ + u_e \]where \(\beta^+\) represents a positron and \(u_e\) is a neutrino.

Understand the Mass Percentages

The given mass percentages are: 18% potassium-40 and 82% argon-40. Assuming all the argon is from the decay, the original amount of potassium-40 would have decayed to argon, and the current potassium-40 is what remains unreacted.

Calculate the Decay Constant

The decay constant (\(\lambda\)) can be calculated using the half-life (\(t_{1/2}\)) formula:\[\lambda = \frac{\ln(2)}{t_{1/2}} = \frac{0.693}{1.2 \times 10^9} \approx 5.775 \times 10^{-10} \text{ yr}^{-1}\]

Determine the Initial and Current Quantity

Let the initial quantity of potassium-40 be \(N_0\), and let the remaining quantity be \(N\). If 18% is \(N\) and 82% has decayed, then \(N/N_0 = 0.18\) and the decayed fraction is 1 - 0.18 = 0.82.

Use the Decay Formula

The formula for radioactive decay is\[N = N_0 e^{-\lambda t}\]Rearranging gives:\[\frac{N}{N_0} = e^{-\lambda t}\]Substituting known values:\[0.18 = e^{-5.775 \times 10^{-10} t}\]

Calculate the Age of the Rock

Take the natural logarithm of both sides to solve for \(t\):\[\ln(0.18) = -5.775 \times 10^{-10} t\]\[t = \frac{\ln(0.18)}{-5.775 \times 10^{-10}} \approx 2.05 \times 10^9 \text{ years}\]

Interpret the Final Result

The age of the moon rock is approximately 2.05 billion years, based on the percentage of potassium-40 remaining and the amount decayed to argon-40.

Key concepts

Potassium-40

Potassium-40 is a naturally occurring isotope of potassium. It is radioactive, which means that it decays over time by emitting particles. This isotope is quite fascinating because it undergoes decay to form different elements. In the case of Potassium-40, it predominantly decays into Argon-40.
This radioactive decay is a random process, but because of its lengthy half-life, it takes a very long time for significant amounts of Potassium-40 to decay. Understanding how Potassium-40 decays is crucial for different scientific applications, including geological dating and understanding the age of rocks and fossils. Knowing its behavior allows scientists to deduce how ancient a geological sample is.
Potassium-40's role in Earth's heat production, due to its radioactive decay, is also an interesting aspect that contributes to our planet's geodynamic processes.

Argon-40

Argon-40 is a product of the decay of Potassium-40. This noble gas is one of the stable isotopes of argon found in Earth's atmosphere. Since Argon-40 accumulates as Potassium-40 decays over time, measuring the amount of Argon-40 relative to the remaining Potassium-40 in a rock sample allows scientists to calculate the age of the rock, a method known as K–Ar dating.
This is particularly useful for dating very old geological formations, such as ancient volcanic rocks on the moon. In the exercise given, almost all the Argon in the moon rock sample results from Potassium-40 decay.

• The more Argon-40 present, the older the rock.
• This makes Argon-40 a key indicator in radiometric dating, providing a window into the past.

Argon-40 is stable, meaning once it is produced, it does not further decay, helping preserve the history of the rock's formation.

Half-life

Half-life is a fundamental concept to understanding radioactive decay. It is the time required for half of the radioactive atoms in a sample to decay. For Potassium-40, this period is exceptionally long at approximately 1.2 billion years.
Knowing the half-life helps scientists predict how long certain isotopes will remain radioactive. It’s also instrumental in calculating the age of rocks. When you determine the ratio of the parent (Potassium-40) to the daughter (Argon-40) isotope, you can estimate how many half-lives have passed and, thus, the age of the rock.
The half-life concept not only applies to dating geological samples but also to understanding processes involved in nuclear physics and materials that use radioactive isotopes.

Beta Decay

Beta decay is the process involved in the transformation of Potassium-40 into Argon-40. During this decay, a beta particle (either an electron or a positron, but in our case, a positron) is emitted. Moreover, a neutrino is also released.
This type of decay is a transformation that changes a neutron in the nucleus into a proton and a beta particle. This process reduces the atomic number of Potassium by one, making it Argon.

• Beta decay is one of the most common decay processes for unstable isotopes.
• It plays a crucial role in changing the identity of elements.

Understanding beta decay is important because it helps explain how certain elements transition to others over geological timescales, and it is a natural phenomenon that influences the chemical composition of rocks and atmospheric gases on Earth.

Decay Constant

The decay constant, denoted as \(\lambda\), is a constant that represents the probability per unit time that a nucleus will decay. It is essential for understanding the rate at which a radioactive isotope decays.
For Potassium-40, the decay constant is calculated using its known half-life. The formula is \(\lambda = \frac{\ln(2)}{t_{1/2}}\). This gives us insight into the predictability and stability of isotopes over time.
Because radioactive decay is a statistical process, with the help of the decay constant, we can forecast how much of a radioactive sample will remain over a given time period. In practical terms, this constant aids geologists to understand isotope ratios and, consequently, the age of materials or formations.